All Quiet on the Exceptional Locus
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911607841357824 |
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| author | Krishna, Ari |
| author_facet | Krishna, Ari |
| contents | We study admissible subcategories of the bounded derived category of a smooth projective surface that are supported on the exceptional locus of a birational morphism. We prove that if $f:X\to Y$ is a birational morphism of smooth projective surfaces, then every admissible subcategory of $D^b(X)$ supported on $\operatorname{Exc}(f)$ is generated by a finite exceptional collection. Moreover, if $K_Y$ is nef, then the same conclusion holds for every admissible subcategory of $D^b(X)$ supported on a proper closed subset of $X$. As a consequence, no nonzero phantom or quasi-phantom subcategory on such a surface can have proper support. The proof combines a splitting lemma for admissible subcategories inside a semiorthogonal decomposition with a single exceptional block, Orlov's blow-up formula, and Pirozhkov's support theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_16277 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | All Quiet on the Exceptional Locus Krishna, Ari Algebraic Geometry 14F08 (Primary) 18G80, 14E05, 14J26 (Secondary) We study admissible subcategories of the bounded derived category of a smooth projective surface that are supported on the exceptional locus of a birational morphism. We prove that if $f:X\to Y$ is a birational morphism of smooth projective surfaces, then every admissible subcategory of $D^b(X)$ supported on $\operatorname{Exc}(f)$ is generated by a finite exceptional collection. Moreover, if $K_Y$ is nef, then the same conclusion holds for every admissible subcategory of $D^b(X)$ supported on a proper closed subset of $X$. As a consequence, no nonzero phantom or quasi-phantom subcategory on such a surface can have proper support. The proof combines a splitting lemma for admissible subcategories inside a semiorthogonal decomposition with a single exceptional block, Orlov's blow-up formula, and Pirozhkov's support theorem. |
| title | All Quiet on the Exceptional Locus |
| topic | Algebraic Geometry 14F08 (Primary) 18G80, 14E05, 14J26 (Secondary) |
| url | https://arxiv.org/abs/2604.16277 |